L6 ECE 5212 Supplementary Notes 10042016 (Summary L5 09302016) F. Jain Absorption coefficient expressions for band-to-band direct and excitonic transitions, indirect transitions, band filling and band tailing.The power continuity equation gives power P at a point in semiconductor medium:-∇ P(x,y,z) = W*ħ (Rate of absorption) (1)The absorption coefficient is also defined in terms of probability of absorption PCV

when an intensity I() is on for t sec duration on a sample of volume V having band-band transitions.

α = ħω * P cv

t * 1I (v )V (2)

PCV is expressed in terms of integral of one transition from state o in valence band to state m in the conduction band.

Pcv= ∫0

∞

Pmo∗ϱ (hv )dv (3)

Here, ϱ is join density of states expressed as: Here, mr is the reduced mass defined below.

ϱ (hv) =[V 4 π (2mr )32 (hv−Eg )

12] /h3

(3b)

1mr

= 1me

+ 1mh (3C)

The probability is obtained by solving a time dependent Schrodinger equation having photon interaction as a perturbation Hamiltonian. It is shown [Moss et al.] that

Pmo(t)=¿Hmo∨¿2¿*sin2 (ω−ωmo )∗t

ħ2¿¿(4)

Here, is the frequency of radiation and mo is the angular frequency corresponding to energy difference Em-Eo.

|Hmo∨¿2¿=2e2 I (v )(|pmo|)

2

3mo2nr ε r cω

2 (5)

Here, pmo is the momentum matrix element which has three components.( px )m0=− j ℏ∫ψm

¿ ∂∂ x ψ 0dτ={ 0 , other than k s

C , other than k=kC=kV (3. 6 )

¿ pmo∨¿2=¿ px,mo∨¿2+¿ py , mo∨¿2+¿ pz ,mo∨¿2¿¿¿¿ p is squareof momentum (6)Add Eq. 7-9

1

Using Eqs. 5, 4 and 3,

Pcv ( t )=2e2 I ( v ) ρ(ωmo)

3m02nr ε 0c ℏ

2ω2 |pmo|2∫

0

∞ sin212 (ω−ωm0 ) t

(ω−ωm0 )2dωm0

(10)

ρ is assumed to be varying much slowly than

sin2 xx2

in the integral, so it is taken out.

Using the standard integral, ∫0

∞ sin2axax 2 dx=πa

2 , [∫0

∞ sin2ax( ax )2

dx=π2 ], or

∫0

∞ sin2 12(ω−ωm0 ) t

(ω−ωm0 )2dωm0

=tLet us take another look at the integration:

∫−∞

∞ t2

4

sin2 12 (ωkm−ω ) t

[ (ωkm−ω) t2 ]2

dω

, let ,

= t2

42t ∫−∞

∞ sin2 xx2 dx

== t

2 ∫−∞

∞ sin2 xx2 dx

=(t/2)*2*t

Pcv( t )=2e2 I( v ) ρ(ωmo) π t

3m02nrε 0c ℏ

2ω2|pmo|

2

(11)E. O. Kane has shown that ¿ pmo∨¿2¿=mo

2P2

ħ2 (12)

Where,P2=( 1

me− 1

mn) 3ħ2Eg(Eg+Δs0)

2(3 Eg+2 Δs0) (13)

P2 = ħ2Eg

2me, if me≪mn Eg>

23Δso (14)

Equation 14 and 12 gives¿ pmo∨¿2

K=Ko¿=mo

2Eg

2me (15)

Substitute equation 15 into equation 5 to obtain Hmo∨¿2¿:

|Hmo∨¿2=2e2 I ( v )

3mo2nr εr c ω

2∗[mo2 Eg

2me]¿ (15b)

Substitute equation 15b into equation 4 to obtain Pmo:

Pmo(t) =2e2 I ( v )

3mo2nr εr c ω

2∗[mo2Eg

2me ] *sin2 (ω−ωmo )∗tħ2¿¿

(15c)

2

Substitute Pmo from Eq. 15C in Eq. 3 to obtain PCV

Pcv (t )=

2e2 I ( v )ϱ (ωmo )πt3m0

2nr ε0 c ħ2ω2 ∗m0

2Eg

2me

(16A)

Subsisting for density of states

Pcv ( t )=

2e2 I ( v )V [4 πh3 ] (2mr )

32 (hv−Eg)

1/2πt

3m02nr ε 0c ħ

2ω2 ∗m02 Eg

2me

(16B)

α=[ħω2 e2[ 4πħ3 ] [2mr]

3 /2

6m02nr ε 0c ħω

2 ][hv−Eg]1 /2[mo

2 Eg

2me ]1/2 (17)

α = A (h – Eg)1/2

Where A= [ ℏω2e2 [ 4 π

ℏ3 ] [2mv ]3

2

6m02nr ε0 c ℏω

2 ][mo2 Eg

2me ]1

2

, (17B)A= 3.38 x 10-7 (nr)-1 (me/mo)1/2 in units of m-1 eV-1/2

Excitonic transitions:For exciton transitions in direct band gap semiconductor, the absorption confident is derived by Elliott (1957).

, γ = ( R

ħω−Eg )1/2

, (Eq. 2 page 164)

Here, R is the exciton binding energy Eex.

It is simplified to as photon energy approaches Eg or below. α ex=A R1 /2π eπγ

(eπγ−e−πγ)/2≈2πA R1 /2 1

(1−e−2 πγ)≈2πA R1/2, at h=Eg. (X,

page 165 Eq. 4B)The plot is shown below (Moss et al.)

3

Excitonic peak at (h-Eg)/R =-1. That is, when h is Eex or R below Eg.

Case II: When photon energy is equal or less than band gap Eg.

,Now we can see from equation (4b):

(6, page 166)We see that Equation (2) expresses both direct band-to-band at energy above band gap and slightly below band gap.Case I:When photon energy is much larger than the band gap hv≫Eg,

α ex=2 A (ħω−Eg)

1 /2 γπ2πγ

, this simplifies to α ex=A (ħω−Eg)1 /2,

which is α ex=α (ħω ) (band-to-band)

Absorption coefficient as a funciton h(without excitonic contributions).

Absorption with excitonic contribution shown in the form of peak below Eg.

4

In the case of quantum wells, the excitonic binding energy Eex or R increases. In addition the direct band-to-band absorption coefficient is higher due to increased joint density of states magnitude.

Calculation of exciton binding energy in bulk layers: (LED Notes, part II)The binding energy of an exciton ranges from 0.004 eV to 0.04 eV. In general the exciton binding energy Eex is:

Eex=mr q

4

8 εo2 εr

2h2(1)

Where:, mr = reduced mass of the exciton given by 1mr

= 1mh

+ 1me (2)

q = electron charge, εr = dielectric constant of semiconductor or carbon nanotube etc.h = Plank's constant

Binding energy can also be viewed as the dissociation energy. The latter being the energy needed to make the electron and hole free, overcoming the Columbic attraction.Quantitatively, Eex = (13./r

2)*(mr/mo); the unit is in electron Volt. This comes from the fact that the

ionization potential of an hydrogen atom is 13.6eV =

moq4

8 εo2 h2

. Semiconductors that have small dielectric constant has larger exciton binding energy.

Bound and free excitons: Excitons are free to move in semiconductor layer or they are bound to certain impurity sites. For example in p-type GaP (dopted with zinc), when electrons are injected they form excitons if GaP

5

is doped with oxygen or nitrogen. In the case of oxygen, Zn-O locations provides sites for excitons to remain localized at those sites as energy levels facilitates exciton formation. Bound exciton at Zn- O site in p-GaP When the electron and hole forming an exciton recombine, this recombination results in photon emission. The probability of photon emission is as high (if not higher) as in the case of direct transitions. The energy of photon hv emitted upon the decay of an exciton at the Zn-O site is:hν=Eg -0 . 3- Eex . (3)if Eex = 0.04 eVhν =2. 24-0 . 3- .04=1 .9eV (4)or wavelength of emission is =1.24/h = 0.65 μm or 6500Å.Bound exciton at nitrogen sites in p-GaPAn injected electron gets trapped at the nitrogen level, and subsequently binds a hole to form an exciton. The decay of this exciton results in a photon emission. The energy of the photon ishν=Eg -0 .08- Eex (6)If Eex = 0.004eVhν =2 . 24- .08- .004=2 .156eV (7)

λ= 1 .242.156

=0 .575 μm=5750 Α(8)

Although GaP is an indirect energy gap material, the electron-hole recombination via excitonic decay is almost like a vertical (direct) transition. This is due to momentum rule relaxation as Heisenberg uncertainty principle (x*p ~h) provides larger latitude in momentum change (p) as x is very small due to bound excitons. Calculation of exciton binding energy in quantum wells: In quantum wells, the heavy and light hole bands split. In unstrained wells and compressive strained wells, the heavy holes are dominant where as in tensile strained quantum wells, light holes are dominant. Exciton binding is higher for smaller width quantum wells than the larger width wells for a given material/semiconductor layer.The electron and hole levels are computed (like HW).

Fig.2(a) Reverse biased p-n diode with MQWs.Change in electro-absorption and exciton peak shift due to E-field:

6

Ee1g

Ehh1

Fig. 1(a) E = 0

The exciton binding energy as well as absorption plot is a funciton of electric field which is applied by an external radio frequency source. One way to apply is to make a p-n junction diode and reverse biasing it. This way the p and n- layers sandwich the multiple quantum well layers.

Electric field moves the electron and holes in different positions. This reduces absorption as well as excitonic binding energy.

Fig. 2(b) Absorption/responsivity as a funciton of applied voltage or electric field.

The electric field also changes the electron E’e1and hole E’hh1energy levels. The change in electron energy levels and change in exciton binding energy E’ex results in the photon energy value at which the exciton absorption peaks. See Fig. 2(b).Note that the band-to-band absorption due to free carriers is not shown in Fig. 2(b). It is to the right of plots shown here. It varies as α = A (h – Eg)1/2.Change in electro-refraction in multiple quantum wells due to E-field:Index of refraction nc is complex when there are losses as photons travel in the medium.

nc= nr – i k Where extinction coefficient is which is k = α/4, here alpha is the absorption coefficient. The square root of permittivity is nc. nc= nr – i k = (ri)1/2

The imaginary part of the permittivity is given by:

The imaginary part i = 2nr , alpha is absorption coefficient, lambda is the wavelength and nr the real part of index of refraction.

i=2nre2/(m2o2)]*(Density of states)*(transition matrix element)*(polarization factor)*(Line shape funciton). Ref. W. Huang, UCONN Ph.D. thesis 1995]

Gaussian Line shape function: L(E)=[1/()1/2]*exp[-(ho-h)2/2], =h/(ln2)-1/2,=1.54*10-13 s.The density of states changes with quantum well, quantum wire and dots. Transition matrix element involves wave functions in one, two or three dimensional confinement.

7

Electron wave function

Hole wave function

Fig. 1(b) Quantum well in the presence of E

E’e1g

E’hh

 

The real part r is related to imaginary part i as

The imaginary and real and index of refraction are:

Use i and relation above to recognize various terms in the following gain expression using excitons: Note gain coefficient g = - (fc + fv -1); here fc and fv are the Fermi distributions for electrons and holes.

gex (ω)= 2π e2

εonrm¿∑ ¿

l,h

[|M b|2⋅2 π1/2|φex (0 )|

2 ¿⋅(|Ψ e( y )Ψ h( y )dy|2|Ψ e( z )Ψ h( z )dz|

2)⋅ρex⋅L(Eex )⋅( f c+ f v -1 ) ]

8

 

 

Indirect band-to-band transitions: summary (pages157, 159-160)

(A) Steps : α (hν )=

ℏωPCV ( t )I (hν )Vt Definition

Initial state ‘0’, intermediate state ‘i’ or ‘i’’, and final state ‘m’. Indirect transition can take place in two ways:

(a)

Process (a) is dominant if The probability that an electron makes a transition from the initial ‘0’ to the final ‘m’ state is

, which is:

(B) Pm0=

4|H i0|2

ℏ4 (ωi0−ω)2|Hmi

± |2⋅sin2 (ωm0−ω±ωp )

t2

(ωm0−ω±ωp )2 (2)Eq.2 is obtained using second order perturbation theory.

Note that Pi0=

|H i 0|2

ℏ2 ⋅sin2 (ωm0−ωi 0)

t2

(ωm0−ωi0 )2 2(b)

(Direct transition)

is the matrix element which represents electron-phonon interaction. ‘+’ sign represents phonon emission and the ‘-‘ is for phonon absorption.

Qualitatively,

is the number of phonons in a particular mode in the crystal of volume V.

By definition, 4) (or the occupancy of the qth mode)

5)The energy balance equations are:

9

(C) is obtained by summing over all pairs of initial states (in the valence band) and all final states (in the conduction band).

7)

PCV ¬PmV ¬Pm0

PmV=∫0

¥

Pm0 ρv (Ev )d (hwm0)and

8) PCV ( t )= ∫

0

hw±E p−Eg

Pmv ρc (Ec )dE c

Where Eq.2, 7 & 8 give:

PCV ( t )=2 p|H i0|

2|Hmi± |2

h2 (DE0−hν )2t ∫

0

hw±Ep−Eg

ρc (Ec ) ρv (Ec−hw±Ep )dE c

Density of states (in volume V)

Conservation of energy – just like joint density of states

Thus, Eq.9 integral is

Or for phonon emission

10

ρc(Ec ) and ρv (Ec−ℏω±E p) in the integrand give

∫Ec1/2Ec

1/2dEc=Ec

2

2≡1

2 (ℏω±E p−Eg )2

(mc≡me¿

mv≡mn¿ )

8V 2mπ3 (mcmv )3/2

h6 (ℏω∓Ep−Eg )2

and Pcv( t )t

=2 π|H i0|

2|Hmi± |2

ℏ2 (ΔE0−ℏω)2⋅8V 2mπ3

h6 (mcmv )3 /2∗(ℏω+Ep−Eg)2

Pcv ( t )t

=2π|H i 0|

2

ℏ2 (ΔE0−ℏω)2⋅BC

V⋅(1

1−e−Ep /kT )⋅8V 2mπ 3

h6

(mcmv )3/2 (ℏω−Ep−Eg )2

Combining Eq.1 with Eq.11, we get 12)

13)

This is:

14)

Exciton transitions in indirect gap semiconductors

Band-to band transitions in heavily doped semiconductors: band filling and band tailing: (174-181)Band filling takes place in low energy gap semiconductors such as InSb where effective mass for electrons and holes is very small as compared with Si and GaAs.

11

Eg+EPEg C

Jα

Eg+EPEg C

Jα

α (hν )=C (ℏω+E p−Eg)2

eE p /kT−1

+C (ℏω−Ep−Eg )

2

1−e−Ep /kT

[Phonon Absorption ] [Phonon Emission ]

C=ℏωI (ν )V

⋅2 π|H i0|

2

ℏ2 ( ΔE0−ℏω )2⋅BC

V⋅8V 2mπ3

h6 (mcmv )3/2

Substiuting for |H i0|2 using Eq. 2(b )

C=ℏωI (ν )V

⋅

2 π {2e2 I ( ν )|Pi0|2

3m02nr ε 0cω

2 }( ΔE0−ℏω )2

⋅BC

V⋅8V 2 Mπ3

h3 (mcmv )3 /2

or C=M (mcmv )3/2e2

6 π2ℏ7m02nr ε0 cω

⋅|Pi0|

2 BC

(ΔE0−ℏω)2

C has contribution from the 0→ i'→m route .

C=M (mcmv )

3/2e2

6π 2ℏ7 m02 nr ε0cω

⋅|Pmi'|

2 BV

(ΔEm−ℏω)2